Scope and Limitations
What the Current Proofs Cover
The current circle kernels use regular symmetric sectors and specific internal and external triangles. Their convergence results assume exact arithmetic and fixed radius.
What Requires Separate Verification
Floating-point and fixed-point stability, computational efficiency, general curves, nonconvex geometry, holes, adaptive sampling, and three-dimensional constructions require their own specifications and evidence.
Refinement
Increasing geometric resolution does not guarantee increasing numerical accuracy indefinitely. Cancellation, quantisation, and underflow or overflow may limit an implementation.
Historical Methods
Cartesian projection and trapezoidal methods have different sampling and endpoint behaviour. Their error rates must not be inferred from the angular kernel proofs.
Research Claims
The existence of a coherent formulation and a valid error proof does not establish original authorship of a local weighting or superiority over alternative implementations.